Relaxation approximation method for the optimal control problem constrained by conservation laws using physics-informed neural networks

We introduce a dual-network physics-informed learning framework designed to solve challenging optimal control problems governed by nonlinear hyperbolic conservation laws. This framework synergistically couples a control variable generator network (CVGN) with an asymptotic-preserving physics-informed neural network (AP-PINN) to jointly discover an optimal control policy and solve the underlying system dynamics. The CVGN parameterizes the control field as a flexible, nonlinear function of space-time coordinates through a neural network. The core innovation lies in the AP-PINN, which addresses the critical challenge of discontinuity in the conservation law solution by solving the corresponding relaxation system instead. This approach transforms the governing equations into a differentiable form amenable to gradient-based optimization. Through a composite loss function, our framework reconciles the optimization objective with physical fidelity without the need for explicit shock tracking. Using the Euler equations as an example of the conservation laws, numerical experiments in one and two dimensions validate the framework's ability to solve complex optimal control problems, including both initial and boundary control scenarios, thereby demonstrating its robustness and versatility.